multivariate distribution function

FFF is non-decreasing in each of its arguments; i.e., for any 1≤i≤n1in1\leq i\leq n, the function Gi:ℝ→[0,1]normal-:subscriptGinormal-→ℝ01G_{i}:\mathbb{R}\to[0,1] given by Gi⁢(x):=F⁢(a1,…,ai-1,x,ai+1,…,an)assignsubscriptGixFsubscripta1normal-…subscriptai1xsubscriptai1normal-…subscriptanG_{i}(x):=F(a_{1},\ldots,a_{{i-1}},x,a_{{i+1}},\ldots,a_{n}) is non-decreasing for any set of aj∈ℝsubscriptajℝa_{j}\in\mathbb{R} such that j≠ijij\neq i.

2.

Gi⁢(-∞)=0subscriptGi0G_{i}(-\infty)=0, where GisubscriptGiG_{i} is defined as above; i.e., the limit of GisubscriptGiG_{i} as x→-∞normal-→xx\to-\infty is 000

3.

F⁢(∞,…,∞)=1Fnormal-…1F(\infty,\ldots,\infty)=1; i.e. the limit of FFF as each of its arguments approaches infinity, is 1.

Generally, right-continuty of FFF in each of its arguments is added as one of the conditions, but it is not assumed here.

If, in the second condition above, we set aj=∞subscriptaja_{j}=\infty for j≠ijij\neq i, then Gi⁢(x)subscriptGixG_{i}(x) is called a (one-dimensional) margin of FFF. Similarly, one defines an mmm-dimensional (m<nmnm<n) margin of FFF by setting n-mnmn-m of the arguments in FFF to ∞\infty. For each m<nmnm<n, there are (nm)binomialnm\binom{n}{m}mmm-dimensional margins of FFF. Each mmm-dimensional margin of a multivariate distribution function is itself a multivariate distribution function. A one-dimensional margin is a distribution function.